Seeing as you haven't really given a really descriptive example of what qualifies "Zen" I'll just come from my own interpretation of the word and what I consider "Zen", being Chan or Zen Buddhism and the things derived from that particular practice, the philosophy and art forms.

Mononoke (Not the movie) would be my favorite and is really the only anime I've seen that I would even consider being into the "Zen" category. I have seen Mushi-shi but I think that's more Shintoism than anything but I've only seen a few episodes and that was like 2 years ago.

It has some influence but I'm not sure to what extent by Chan Buddhism or Zen Buddhism as it's known in Japan. From what I remember, it draws a lot of it's story from Japanese folklore, theme from more folklore but a bit from Zen Buddhism...probably in the way Mononoke manifest and the way in which they relate to humans and the world...maybe. The protagonist is probably practicing more esoteric practiced arts of Zen Buddhism unless that's some Shinto stuff, if anything it's probably a loose amalgamation of both. The show doesn't really dive into the philosophy of Zen Buddhism, but it does view the world in similar but very rudimentary way from what I've gathered in my limited personal studies. So it fits my personal interpretation of Zen but not by a whole lot.

Mononoke is pretty dope for a lot of reasons, the art is really appealing to me and quite unique and fitting to the whole atmosphere and it's not only appealing to eye it goes beyond that, I like the protagonist since he doesn't meddle too much but in a sense he is also an observer of the story like the viewers (kinda like he's detached but involved all at the same time), the stories are very nice renditions of Japanese folklore, and I feel like the whole series is really coherent and well made in all aspects. You could watch multiple times and still see new things.

Haibane is the most profound, Mushishi the most calming, Aria the most charming, Kino the most allegorical, Mononoke the most esoteric, Natsume the most healing and one of my personal favorites: Mouryou no Hako the most surreal.

STILL NEED TO SEE SOUND OF THE SKY THO. It's amazing I've let it go un-watched for this long.